Ch-6: THE TRIANGLE AND ITS PROPERTIES Worksheet No. 12 Q1. Choose the correct option: 1. In a triangle, a line segment joining any vertex to the mid-point of the opposite side is called a/an: a) Altitude b) Perpendicular c) Median d) Altitude 2. A triangle whose two angles measure \( 30^{\circ} \) and \( 120^{\circ} \) is: a) Scalene b) Isosceles c) Equilateral d) Right Angled 3. In a right-angled triangle, the lengths of two legs are 6 cm and 8 cm . The length of the hypotenuse is: a) 14 cm c) 10 cm b) 11 cm d) 12 cm 4. If \( p, q, r \) are the lengths of the three sides of a triangle, then which of the following statements are correct? a) \( p+q=r \) c) \( p+q>r \) b) \( p+qr \) 5. Point of concurrence of the altitudes of a triangle is called a) Centroid c) Incentre b) Orthocentre d) Circumcentre Q2. Fill in the blanks: 1. In an Equilateral triangle, each angle is equal to \( \qquad \) degrees. 2. A triangle cannot have more than \( \qquad \) right angle (s). 3. The longest side of a right-angled triangle is the \( \qquad \) . 4. Dimensions of a rectangle are 15 cm and 8 cm ; The length of its diagonal is \( \qquad \) cm. 5. In a triangle, an exterior angle is \( 135^{\circ} \). If the measure of one interior opposite angle is \( 45^{\circ} \) then measure of the other interior opposite angle is \( \qquad \) Q3. State True / False: 1. An equilateral triangle is an Isosceles triangle. \( \qquad \) 2. A triangle can have sides \( 6 \mathrm{~cm}, 4 \mathrm{~cm}, 10 \mathrm{~cm} \). \( \qquad \) 3. In an isosceles triangle, two angles are equal. \( \qquad \) 4. Meeting point of the medians of a triangle is Centroid. \( \qquad \) 5. Altitudes of a triangle are always in the interior of the triangle. \( \qquad \)

### **Worksheet No. 12: THE TRIANGLE AND ITS PROPERTIES** --- #### **Q1. Choose the correct option:** 1. **In a triangle, a line segment joining any vertex to the mid-point of the opposite side is called a/an:** **c) Median** 2. **A triangle whose two angles measure \( 30^{\circ} \) and \( 120^{\circ} \) is:** **b) Isosceles** 3. **In a right-angled triangle, the lengths of two legs are 6 cm and 8 cm. The length of the hypotenuse is:** **c) 10 cm** 4. **If \( p, q, r \) are the lengths of the three sides of a triangle, then which of the following statements are correct?** **c) \( p + q > r \)** 5. **Point of concurrence of the altitudes of a triangle is called** **b) Orthocentre** --- #### **Q2. Fill in the blanks:** 1. **In an Equilateral triangle, each angle is equal to** **60** **degrees.** 2. **A triangle cannot have more than** **one** **right angle(s).** 3. **The longest side of a right-angled triangle is the** **hypotenuse** **.** 4. **Dimensions of a rectangle are 15 cm and 8 cm; The length of its diagonal is** **17** **cm.** 5. **In a triangle, an exterior angle is \(135^{\circ}\). If the measure of one interior opposite angle is \(45^{\circ}\) then measure of the other interior opposite angle is** **90** **degrees.** --- #### **Q3. State True / False:** 1. **An equilateral triangle is an Isosceles triangle.** **True** 2. **A triangle can have sides \(6~\text{cm}\), \(4~\text{cm}\), \(10~\text{cm}\).** **False** 3. **In an isosceles triangle, two angles are equal.** **True** 4. **Meeting point of the medians of a triangle is Centroid.** **True** 5. **Altitudes of a triangle are always in the interior of the triangle.** **False** --- ### **Detailed Solutions:** #### **Q1. Multiple Choice Questions** 1. **Median**: A median connects a vertex to the midpoint of the opposite side. 2. **Isosceles**: Two equal angles (\(30^{\circ}\) each) imply two equal sides. 3. **Hypotenuse Calculation**: \( \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm} \). 4. **Triangle Inequality**: The sum of any two sides must be greater than the third side. 5. **Orthocentre**: The intersection point of all three altitudes. #### **Q2. Fill in the Blanks** 1. **60 degrees**: All angles in an equilateral triangle are equal and sum to \(180^{\circ}\). 2. **One**: A triangle cannot have two right angles as their sum would exceed \(180^{\circ}\). 3. **Hypotenuse**: The side opposite the right angle in a right-angled triangle. 4. **17 cm**: Diagonal \( = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17 \text{ cm} \). 5. **90 degrees**: Exterior angle \( = 135^{\circ} \), one opposite interior angle \( = 45^{\circ} \), so the other is \(135 - 45 = 90^{\circ}\). #### **Q3. True/False Statements** 1. **True**: An equilateral triangle has all sides equal, which satisfies the condition for being isosceles. 2. **False**: The sides \(6\), \(4\), and \(10\) do not satisfy \(6 + 4 > 10\). 3. **True**: By definition, an isosceles triangle has at least two equal angles. 4. **True**: The centroid is where the three medians intersect. 5. **False**: In an obtuse triangle, some altitudes lie outside the triangle. --- Feel free to reach out if you have any further questions or need more explanations!